Coordination and Culture

Jean-Paul Carvalho∗

Department of EconomicsUniversity of California, Irvine

Abstract

Culture constrains individual choice, rendering certain actions impermissible ortaboo. While cultural constraints may regulate behavior within a group, theycan have a pernicious effect in multicultural societies, inhibiting the emergenceof unified social conventions. We analyze interactions between members of twocultural groups who are matched to play a coordination game with an arbitrarynumber of actions. Due to cultural constraints, miscoordination prevails despitestrong incentives to coordinate behavior. In an application to identity-based con-flict, exclusive ethnic and religious identities persist in poorer and more unequalsocieties. Occasional violation of cultural constraints can make miscoordinationeven more stable.

This version: 19 February 2016

JEL Classification: C72, C73, Z1

Keywords: Culture, Conflict, Coordination Failure, Stochastic Stability

∗This paper has benefitted greatly from advice by Peyton Young, as well as comments by the AssociateEditor, two anonymous referees, Blake Allison, Ken Binmore, Rob Boyd, Michael Caldara, David Myatt,Tom Norman, Michael Sacks, Stergios Skaperdas, Christopher Wallace and seminar participants at UCLAAnderson, Claremont Graduate University, the Institute for Mathematical Behavioral Sciences, UC Irvineand the University of Western Australia. All errors are mine. Financial support from the CommonwealthBank Foundation in the form of a John Monash scholarship is gratefully acknowledged.

1 Introduction

Culture constrains individual choice. In matters of diet, dress, language, manners and much

else, one’s culture renders certain choices impermissible or taboo.1 There are many well

known taboos. Vegetarians do not consider switching to meat as circumstances dictate. It

remains taboo for the Amish to use modern technology, even as it becomes more attractive.

Christians in the Middle Ages observed usury prohibitions. Moreover, there are many less

obvious ways in which culture can restrict an individual’s choice set by making certain types

of behavior unthinkable, unpalatable or prohibitively costly to learn. For example, suppose

that members of two different religious groups can form new business contacts by interacting

socially. Despite gains from social coordination, no individual would consider meeting at the

other group’s place of worship. To interact, they must coordinate on a neutral venue.

This paper proposes a particularly simple notion of culture. Let X be the global choice

set. A member of cultural group k chooses from the set Xk ⊆ X. This is what Sen (1977)

refers to as “commitments”, Harsanyi (1982) as “moral values” and Rao & Walton (2004)

as “constraining preferences”.2 It then explores the consequences of this notion of culture.

While cultural constraints can help to coordinate and regulate behavior within a group, it

is possible that they have a pernicious effect in multicultural societies, inhibiting the emer-

gence of unified social conventions. This could occur, we suggest, even when coordination is

attainable, that is when XA ∩XB is nonempty for some groups A and B.

Consider a set of boundedly rational agents partitioned into two cultural groups, each with

its own cultural constraint. Individuals are matched recurrently with members of the other

group to play a coordination game with an arbitrary number of actions. They adapt their

choices over time and occasionally make mistakes, a la Kandori, Mailath & Rob (1993) and

Young (1993a). We are interested in whether they can learn to coordinate with each other

within cultural constraints. The results are rather pessimistic. Certain forms of cultural

1For example, Tetlock et al. (2000) find that experimental subjects express moral outrage at even contem-plating certain taboo transactions, including buying and selling of human body parts for medical transplantoperations, surrogate motherhood contracts, adoption rights for orphans, votes in elections for political of-fice, the right to become a U.S. citizen and sexual favors. See Roth (2007) for an account of how repugnanceconstrains markets.

2See also Henrich et al. (2001) who suggest that culture “limits choice sets” [p. 357]. Conventionaldefinitions in economics equate culture with shared preferences (e.g. Bisin & Verdier 2000) and strategicbeliefs (e.g. Greif 1993).

1

diversity—combinations of cultural constraints and preferences—can lead to the breakdown

of coordination across groups, even where coordination is attainable and Pareto dominates

miscoordination. Nevertheless, there are multiple equilibria and the emergence of a unified

social convention across groups is always possible in the long run (as long as XA ∩ XB is

nonempty).

To select among equilibria, we study the evolution of play when the error rate is positive

but vanishingly small (Foster & Young 1990). Since every state of coordination Pareto

dominates every state of miscoordination, one might expect miscoordination to be a ten-

uous phenomenon. On the contrary, miscoordination is stochastically stable for an open

set of parameters.3 In particular, we employ a graph-theoretic argument to show that if

miscoordination pairwise (strictly) risk dominates all other equilibria, then it is the unique

stochastically stable class. Because we study large coordination games, this does not follow

from existing results, nor is the proof trivial. In addition, we derive a condition under which

pairwise risk dominance is necessary and sufficient for miscoordination to be stochastically

stable.

The analysis is applied to an example of identity-based conflict in which it is Pareto efficient

for members of two ethnic groups to coordinate on an inclusive (e.g. national) identity. Sen

(2006) argues that the roots of identity-based violence lie in the emphasis on an exclusive

sense of self.4 He claims that through reason one can choose to arrive at a more inclusive

social identity. As we show, however, even when strong incentives to coordinate on an

inclusive identity exist, exclusive (e.g. ethnic) identities may persist due to evolutionary

pressures. In particular, we demonstrate that exclusive ethnic and religious identities are

more likely to persist in poorer and more unequal societies.

A question remains as to whether occasional violation of cultural constraints could destabilize

3This phenomenon is qualitatively different to the selection of a Pareto-inefficient coordination equilib-rium when agents use the same action set (Kandori, Mailath & Rob 1993, Young 1993a). Here, play mightnot settle into any coordination equilibrium, even though every coordination equilibrium Pareto dominatesmiscoordination.

4Important contributions to understanding identity-based conflict and cooperation are also made byFearon & Laitin (1996), Laitin (2007), Esteban & Ray (2008), McBride et al. (2011), Greif & Tabellini(2012), Jha (2013) and Sambanis & Shayo (2013). Sambanis and Shayo present a formal theory in whichindividuals either identify with their ethnic group or the nation. Under certain conditions, multiple equilibriaexist and cascades of ethnic identification can occur (see also Kuran 1998). Such miscoordination betweencultural groups, as we show, is a more general phenomenon, not limited to ethnic conflict and independentof specific feedback mechanisms. By analyzing an explicit out-of-equilibrium process, we are also able toselect among different equilibria and show that miscoordination is a surprisingly stable outcome.

2

miscoordination. Since miscoordination is supported as an equilibrium only when cultural

constraints differ in a particular way, one might expect violation of cultural constraints to

weaken the stability of miscoordination or at least limit what can be said. We find otherwise.

Not only are the results sharper, but moreover small-scale violations can strengthen misco-

ordination, expanding the set of parameters for which it is stochastically stable. Specifically,

we derive a necessary and sufficient condition for miscoordination to be stochastically stable,

which is weaker than risk dominance.

This paper is related to several different lines of work. Firstly, our results contribute to the

study of stochastic stability in general asymmetric coordination games. Early work analyzed

stochastic stability in symmetric 2× 2 coordination games (e.g. Young 1993a, 1996, Ellison

1993),5 showing that risk dominance is necessary and sufficient for stochastic stability. Prior

work on asymmetric coordination games has largely focussed on deterministic dynamics

(Samuelson & Zhang 1992) and two-action games (Staudigl 2012).

Young (1993a, p. 73) shows by example that beyond 2×2 coordination games, risk dominance

need not be sufficient for stochastic stability. Indeed risk dominance is a pairwise concept,

determining the likelihood of direct transitions between equilibria. Hence the logic from

2 × 2 coordination games does not naturally extend to larger games. However, Young’s

example does not match our payoff structure. Kandori & Rob (1993) show that, under

certain conditions, an equilibrium is stochastically stable if it pairwise risk dominates every

other. Their result relies upon a ‘total bandwagon property’, the violation of which is critical

to our analysis (as demonstrated later). Hence our results do not follow from prior work.

Ellison (2000) shows, for symmetric L × L coordination games, that when a half-dominant

equilibrium exists, it is stochastically stable. Our sufficient condition for stochastic stability

is substantially weaker than half dominance. We also identify a condition under which risk

dominance is necessary and sufficient for stochastic stability. In addition, when individuals

occasionally violate cultural constraints, we derive a necessary and sufficient condition that

is weaker than risk dominance.

Miscoordination has been studied from a different perspective in 2× 2 coordination games.

Myatt & Wallace (2004) analyze 2 × 2 coordination games in which agents’ payoffs are de-

termined by random draws from a normal distribution. Miscoordination occurs in certain

5A notable exception is Young (1993b) who examines bargaining between members of heterogeneousgroups.

3

matches. As payoff trembles vanish, however, a single pure equilibrium is selected and coor-

dination is almost always achieved. Quilter (2007) and Neary (2012) study 2×2 coordination

games with nonvanishing differences in payoffs. Players belong to one of two groups and play

a single action against all members of the population. When group A members prefer to

coordinate on action 1 and group B members on action 2, there is a tradeoff between ingroup

coordination on one’s preferred action and coordination with outgroup members. As a re-

sult, multiple conventions can coexist.6 Because our focus is on between-group interactions,

the type of miscoordination that we uncover can occur where there is no tradeoff between

ingroup and outgroup coordination. As such, our results also apply when individuals are

flexible, viz. able to choose one action when interacting with ingroup members and another

action with outgroup members.7

Among other related work, Kuran & Sandholm (2008) study culture as an evolving dis-

tribution of preferences and behaviors, not a cultural constraint on behavior. There is no

breakdown of Pareto-efficient coordination in their work and cultural convergence eventu-

ally occurs among different groups. Benabou & Tirole (2011) and Fershtman et al. (2011)

analyze the conditions under which taboos emerge. Benabou and Tirole present a theory of

self-signaling in which taboos protect one’s sense of self or identity. Fershtman et al. propose

that taboos provide public benefits and examine the conditions under which they can unravel

through a series of deviations. Our focus is on whether individuals from different cultural

groups can learn to coordinate within a fixed set of cultural constraints or taboos.

The paper is structured in the following manner. The model is introduced and analyzed in

section 2. Section 3 studies an application to identity choice and an extension to small-scale

violations of cultural constraints. Section 4 concludes. All proofs and technical lemmas are

located in the Appendix.

2 The Model

In this section, we analyze a two-population model of social coordination. Interactions occur

between groups and individuals are bound by cultural constraints.

6In addition, when agents are differentially located in space, different groups can settle on differentconventions (e.g. Young & Burke 2001).

7Members of ethnic or religious minorities often adopt different modes of behavior when interactingoutside of their communities (see Akerlof & Kranton 2000, p. 738-9).

4

2.1 The Underlying Game

Consider a society composed of a finite set of roles N , with typical members i and j. A

(possibly) changing cast of players fill these roles.8 For convenience, we refer to elements of

N as players (rather than roles).

Each player belongs to one of two equally sized cultural groups k ∈ {A,B}. The set of

players comprising group k is denoted by Nk, where |Nk| = n is finite. Each group has a

different culture, which proscribes certain kinds of behavior.

Individuals are matched with a player drawn uniformly at random from the other group to

play a coordination game.9 The finite set X is the set of all actions that can potentially

be chosen in the game. For example, X can represent a menu of diets, languages or social

identities.

In addition to coordination, individuals have two considerations when choosing an action.

The first consideration is respect for cultural constraints. This is the key assumption:

Condition 1. (Cultural Constraints) A group k member chooses from the set of actions

that are culturally permissible, Xk ⊆ X.

Such a restriction on an agent’s strategy set might come about through the internalization

of culturally accepted standards of behavior during socialization or as a result of sanctions

imposed by the group, though we do not model this process here (see Frank 1988, Elster 1989,

Akerlof & Kranton 2000). For example, a Turkish woman raised in a secular household may

not consider wearing a headscarf, whereas a Turkish woman raised in a religious household

may not consider going out in public without one.10 In section 3.2, we consider a setting in

which all actions are available to all players but each action in X\Xk is strictly dominated

for k members by some action in Xk.

Secondly, they have culturally defined preferences over actions that are independent of their

partner’s action. The payoff to a group k member who is matched with player j and plays

8Over time, we can think of players “dying” and their roles being filled by incoming players who inherittheir predecessor’s culture. One such example is the vertical transmission of culture from parent to child.

9The results in this section hold if we assume that in every period each agent is matched with everyagent from the other group, with a set Rt of individuals selected to revise their strategies as above. Theapplication analyzed in section 3.1, however, is developed for pairwise interactions.

10For theories of veiling see Patel (2012) and Carvalho (2013).

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x is given by

uk(x, xj) = I(x, xj)θk + δkx. (1)

The first term on the right-hand side (RHS) of (1) is defined as follows: I(x, xj) = 1 if x = xj

and zero otherwise. Hence θk > 0 is the payoff to a group k member from coordination. The

second term, δkx, represents the cultural payoff from choosing action x for a group k member.

Myatt & Wallace (2004) were the first to introduce such payoff disturbances in the context

of coordination games. We refer to an action x ∈ argmaxx∈Xkδkx as group k’s cultural ideal.

Let dk ≡ maxx∈Xk\Xk′δkx, k 6= k′, denote the maximal payoff to a group k member from

choosing a mutually impermissible action. We shall refer to dk − δkx as group k’s cultural

bias against x.

We impose the following non-degeneracy condition on payoffs:

Condition 2. (ND) δkx′ − δkx < θk for all (x, x′) ∈ X2 and k = A,B.

In this paper, we are interested in social coordination. When ND is satisfied, coordination

on any mutually permissible action Pareto dominates miscoordination despite conflicting

cultural ideals. When ND is not satisfied, some mutually permissible actions are weakly

dominated.

2.2 Adaptive Choice and Population Dynamics

Interactions take place in discrete time, indexed by t = 0, 1, 2, . . . Let ztk ≡(xti)i∈Nk

be

the group-k action profile at the beginning of time t. The state in period t is defined by

the action profiles of the two groups, zt ≡(ztA, z

tB

). The associated (finite) state space is

Z = XnA ×Xn

B. The process begins in an arbitrary state z0 ∈ Z.

In each period t ≥ 1, a set of players Rt is selected at random. Each player i ∈ Rt gets the

opportunity to revise his strategy in period t. Let f(R) denote the probability that set R is

selected in a given period. We assume that f(R) > 0 for all R ∈ P(N), the power set of N .

(Hence, among other things, simultaneous revisions occur with positive probability.)

Each revising k member at time t plays a myopic and noisy best response to ztk′ . Let pk′x be

the proportion of group k′ members playing x. With high probability 1−ε, a revising player

6

chooses a constrained best response by maximizing pk′xθk + δkx, subject to x ∈ Xk. (When

there are ties, each best response is played with equal probability.) With low probability ε,

a revising group k member instead selects an action from Xk at random. Nonbest responses

represent mistakes and/or random experimentation.

The best response protocol with uniform errors (see Kandori, Mailath and Rob 1993, Young

1993a) implies myopia on the part of agents in that choices are based solely on the current

state, a standard assumption in evolutionary game theory (see Young 1998, Sandholm 2010).

Computing the effect of current choices on the future trajectory of play is a complex and

costly exercise. Myopic behavior is thus natural in large populations of boundedly rational

individuals.

It is well known that this revision protocol produces a particular kind of evolutionary dynamic

at the population level. There exist time-homogeneous transition probabilities between all

pairs of states z, z′, denoted by Pzz′ . They define a finite Markov chain with a |Z| × |Z|transition probability matrix which depends on the noise level ε. For ε > 0, all pairs of

states communicate, so the Markov chain is irreducible and thus ergodic—it has a unique

stationary distribution µε that is independent of the initial state z0.11 The process is also

aperiodic when ε > 0, since there is a positive probability of remaining in any given state. As

such, the stationary distribution tells us a lot about the asymptotic behavior of the process.

Not only does the relative frequency of visits to state z up through time t converge to the

frequency given by the unique stationary distribution µε, but so does the probability of being

in state z at time t.

Typically, we can greatly reduce the set of states in the support of µε by taking the noise

level to zero. We rely on the following equilibrium concept due to Foster & Young (1990):

Stochastic Stability. A state is stochastically stable if it is in the support of µ = limε→0µε.

A stochastically stable class is a recurrent class composed of stochastically stable states.12

As the noise level becomes arbitrarily small, the process spends virtually all of the time as

t→∞ in the stochastically stable classes.

11Recall that state z′ is accessible from z if there exists a positive probability path from z to z′. Thestates communicate if they are each accessible from the other.

12A set of states E is closed if for all x ∈ E and y /∈ E, Pxy = 0. A recurrent class is a closed communicationclass.

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2.3 Coordination and Miscoordination

Let us begin by studying the asymptotic behavior of the best response dynamic with ε = 0,

before proceeding to the stochastic stability analysis. We would like to know if members of

different cultural groups can learn to coordinate with each other. There are two types of

recurrent states (i.e. members of a recurrent class):

Definition 1. (Coordination) A state of coordination is one in which xi = x for all i ∈ N .

Definition 2. (Miscoordination) A state of miscoordination is one in which xi 6= xj whenever

i ∈ Nk and j ∈ Nk′ , k 6= k′.

In a moment, we will show that social coordination can emerge even when payoffs differ

across groups. Differing cultural constraints, however, create a new possibility in which

social coordination permanently breaks down. Let us define the following relation:

Definition 3. (Compatibility) Cultures A and B are compatible if XA ∩ XB is nonempty.

They are incompatible otherwise.

Clearly coordination between incompatible cultures is impossible. It turns out, however,

that miscoordination can arise even among compatible cultures, when cultural ideals differ

in a particular way. Let Xk ≡ argmaxx∈Xkδkx be the set of cultural ideals for group k.

Definition 4. (Misalignment) Cultures A and B are misaligned if XA ∩ XB = ∅ and

XB ∩XA = ∅. They are aligned otherwise.

Two cultures are misaligned when every ideal action for one group is impermissible for the

other. If the cultures are incompatible, then they are also misaligned, since XA ⊆ XA

and XB ⊆ XB. The converse is not true; unlike compatibility, alignment depends on the

difference between the cultural ideals of each group. Compatible cultures are misaligned if

and only if they both have a positive cultural bias against all mutually permissible actions,

i.e. dk − δkx > 0 for all x ∈ XA ∩XB and k = A,B.

Define M = XnA × Xn

B as the set of states in which everyone plays a culturally ideal action.

When the cultures are misaligned each z ∈M is a state of miscoordination. In this case, we

simply refer to M as miscoordination. We can state the following result:

8

Proposition 1 The best response dynamic (ε = 0) converges almost surely to:

(i) a state of coordination if the cultures are aligned,

(ii) either a state of coordination or miscoordination M, depending on the initial state z0,

if the cultures are compatible but misaligned,

(iii) miscoordination M if the cultures are incompatible.

To prove Proposition 1, we identify the conditions under which the states of coordination

and M are recurrent classes of the best response dynamic operating on this game and

hence correspond to its static Nash equilibria. In addition, we show that these are the only

recurrent classes. As any finite Markov chain converges almost surely to one of its recurrent

classes, this suffices to establish the proposition. All proofs are in the Appendix.

When cultures are aligned, individuals inevitably learn to coordinate with each other in cross-

cultural interactions. When cultures are incompatible, they cannot. The interesting case is

when cultures are compatible but misaligned. In this case, coordination between different

cultural groups can permanently break down even though coordination Pareto dominates

miscoordination. Whether players can learn to coordinate depends on initial conditions. If

play begins with a sufficiently large proportion of individuals choosing a mutually permissible

action, x ∈ XA ∩ XB, then the prospect of coordination will induce revising agents to

depart from their cultural ideals. Otherwise, miscoordination can arise.13 Suppose that the

process begins in a state of miscoordination. A revising player believes he has no chance of

coordinating with members of the other group, so he chooses one of his ideal actions, say

x. If the cultures are misaligned then x /∈ XA ∩ XB, so the dynamic remains in a state of

miscoordination. If however the cultures are aligned, then revising players from at least one

group will respond with a mutually permissible action. Thus the dynamic can exit from a

state of miscoordination and social coordination may be achieved.

It is cultural diversity that hinders social coordination, not any other aspect of our setup.14

13This does not mean that the state space can be partitioned into a basin of attraction for each state ofcoordination and a basin of attraction forM. There are some states from which both a state of coordinationand miscoordination can be reached with positive probability, depending on which players are chosen torevise their strategy.

14This accords with empirical evidence. For example, interracial, interethnic and interreligious marriagesare more likely to end in divorce than homogamous marriages (see Fu (2006) and references therein) and

9

When players are homogeneous (a single-population model), standard results apply (e.g.

Young 1998, p. 68): the best response dynamic converges almost surely to a state of co-

ordination from any initial state. In our two-population model, the difference in cultural

constraints is critical to social miscoordination. Notice that for two cultures to be compat-

ible and misaligned, there must be an action that group A members can take but group

B members cannot and an action that group B members can take but group A members

cannot. Suppose instead that XA ⊆ XB or XB ⊆ XA. In this case, the cultures are aligned

and, by Proposition 1(i), social coordination emerges in the long run between the two groups

regardless of differences in their cultural ideals.

2.4 Stochastically Stable Miscoordination

Henceforth we shall examine interactions between groups that are compatible but misaligned.

When the cultures are incompatible, we know thatM is the unique recurrent class of the best

response dynamic (ε = 0). Since the set of stochastically stable states is a non-empty subset

of the set of recurrent classes of the unperturbed dynamic, M is the unique stochastically

stable set of states.

As long as the cultures are compatible, the states of coordination are recurrent classes of

the evolutionary dynamic. Moreover, each state of coordination Pareto dominates every

state of miscoordination. Hence one might expect social miscoordination to be a tenuous

phenomenon, accessible from only a small set of initial conditions. In this section, we show

that, on the contrary, miscoordination is the most likely outcome of play in the long run for

an open set of parameters. In fact, we can be more precise.

Consider the perturbed best response dynamic with error rate ε > 0. Again, studying

stochastic stability by taking the limit ε→ 0 will allow us to make sharp statements about

the asymptotic behavior of the evolutionary dynamic, independently of initial conditions.

As discussed in the introduction, not much is known about stochastic stability in asymmetric

coordination games with more than two actions. In our setting, individuals from two different

populations play a coordination game with an arbitrary number of actions. Interactions occur

cultural differences reduce the volume of and returns from cross-border mergers (Ahern et al. forthcoming).The subdiscipline of cross-cultural management is devoted to such issues. See Akerlof & Kranton (2010) fornumerous examples of organizational conflict along gender, class and racial lines. On the other hand, Page(2007) shows how organizations can benefit from diversity.

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between populations, with each population having different idiosyncratic payoffs over actions

and different action sets. We shall now show that risk dominance is sufficient for M to be

stochastically stable. We also derive a condition under which it is necessary and sufficient.

Define group k’s opposition to coordination on x by Dkx ≡ (dk− δkx)/θk. This is the group’s

cultural bias against x scaled by its coordination payoff.

According to Harsanyi & Selten (1988), an equilibrium z∗ strictly risk dominates another

z∗∗ if the product of (unilateral) deviation losses from z∗ is greater than that from z∗∗. An

equilibrium is strictly risk dominant if it pairwise strictly risk dominates all other equilibria.

In our setting, miscoordination M strictly risk dominates a state of coordination on x ∈XA ∩XB if

(dA − δAx)(dB − δBx) > (θA + δAx − dA)(θB + δBx − dB)

θB(dA − δAx) + θA(dB − δBx) > θAθB

DAx +DBx > 1.

Hence:

Definition 5. Miscoordination M is strictly risk dominant if

minx∈XA∩XB

(DAx +DBx

)> 1. (2)

Let us also define the notion of simple payoffs, in which cultural payoffs are uniform over

mutually permissible actions:

Definition 6. Payoffs are simple if δkx = δk for all x ∈ XA ∩XB and k = A,B.

We can now state the result.

Proposition 2 Suppose the groups are compatible but misaligned. Consider the perturbed

best response dynamic in the small noise limit ε→ 0, for a sufficiently large population size

n.

11

(i) Miscoordination M is the unique stochastically stable class if it is strictly risk domi-

nant.

(ii) When payoffs are simple, M is the unique stochastically stable class if and only if it is

strictly risk dominant.

The proof of Proposition 2 employs a graph-theoretic method developed by Young (1993a).

It relies on several technical lemmas (stated and proved in the appendix), as well as two

tree-surgery arguments. Proposition 2(i) states that strict risk dominance is sufficient forMto be stochastically stable.15 Proposition 2(ii) states that when cultural payoffs are uniform

over mutually permissible actions, this condition is necessary and sufficient.

Before relating this result to prior work, several further remarks are in order.

Remark 1. The condition for strict risk dominance ofM, (2), is stronger than misalignment.

Recall that M is a recurrent class of the unperturbed best response dynamic if and only if

the cultures are misaligned. Misalignment is equivalent to

minx∈XA∩XB

mink∈{A,B}

Dkx > 0. (3)

This does not preclude

minx∈XA∩XB

maxk∈{A,B}

Dkx <1

2,

in which case condition (2) is violated.

In addition, (2) can only be satisfied when the cultures are misaligned. Suppose contrary to

(3) that Dkx ≤ 0 for some k ∈ {A,B} and x ∈ XA ∩ XB, so that the cultures are aligned.

In this case,

DAx +DBx ≤ maxk∈{A,B}

Dkx. (4)

Recall that condition ND requires θk > dk− δkx for all x ∈ XA∩XB and k ∈ {A,B}. Hence

the RHS of (4) is less than one, which in turn means that (2) is violated.

15The population size n needs to be sufficiently large only to break ties in resistances between recurrentclasses induced by rounding up to the nearest integer. When n is small, multiple classes can be selected.But as long as M is stochastically stable for some population size n it remains stochastically stable for allsmaller n.

12

How much stronger than misalignment is the condition for stochastic stability (2)? To

illustrate, consider the following symmetric example.

Example 1. θk = θ, dk = d and δkx = δ for all x ∈ XA ∩XB and k = A,B.

By ND, d − δ < θ. Misalignment requires that d − δ > 0. The condition for stochastic

stability reduces to d−δ > 12θ. ForM to be uniquely stochastically stable then, the cultural

bias need only be greater than half the coordination payoff. Miscoordination can prevail even

when incentives for coordination are much stronger than cultural incentives.

The intuition is as follows. The mutually permissible actions yield high payoffs when everyone

coordinates on them. Whether the gain from coordination is realized, however, depends on

the actions of the other group. Cultural payoffs, in contrast, are payoffs to taking actions

per se, independently of actions taken by the other group. If cultural payoffs favor mutually

impermissible actions, then states of miscoordination may be more robust to random shocks,

yielding higher payoffs out of equilibrium than the states of coordination. Out-of-equilibrium

behavior is important in our context because the perturbed dynamic can transit between

recurrent classes due to a sequence of errors. In the limit as ε → 0, the dynamic spends

virtually all of the time in the classes that require few errors to get into and many errors to

get out of. Thus, when the cultural payoffs to mutually impermissible actions are relatively

high, coordination is easily destabilized and miscoordination takes place virtually all of the

time.

This mirrors the logic behind selection of risk-dominant equilibria in 2 × 2 coordination

games (Kandori et al. 1993, Young 1993a) and larger coordination games satisfying certain

properties (Kandori & Rob 1993), as well as half-dominant equilibria in symmetric coordi-

nation games (Ellison 2000). Nevertheless, we shall now establish that Proposition 2 does

not follow from existing results. (Nor is the proof trivial.) Hence it is not obvious that the

intuition derived from other settings should apply here.16

Young (1993a, p. 73) shows by example that beyond 2×2 coordination games, risk dominance

need not be sufficient for stochastic stability. His example does not, however, match our

payoff structure. In fact, Kandori & Rob (1993) show that, under certain assumptions, an

16Naturally, the results coincide when |XA ∩XB | = 1, but our analysis is more general.

13

equilibrium is stochastically stable if it pairwise risk dominates every other. Their result

relies on a ‘total bandwagon property’ being satisfied, which in our setting means:

pθk + δkx > δkx′ for all p > 0, (x, x′) ∈ X2 and k = A,B,

or equivalently δkx = δkx′ for all (x, x′) ∈ X2 and k = A,B. This in turn implies that

the cultures are aligned, so that M is not a candidate for stochastic stability. Hence the

violation of one of their key assumptions is critical to our analysis.

The other result that we have to consider is the selection of half-dominant equilibria in

symmetric coordination games by Ellison (2000). Our condition for M to be stochastically

stable is weaker than half-dominance. According to Morris et al. (1995), a strategy x is

half-dominant if it is a strict best response to any state in which at least 12

of the population

play x. This is a symmetric notion, that does not account for the potential asymmetries

in multi-population games. In our asymmetric setting, with between-group interactions,

half-dominance can be defined as follows (see Kajii & Morris 1997):17

Definition 7. (Half-Dominance) Consider a state z in which, for each group k ∈ {A,B},at least 1

2n players choose a mutually impermissible action x ∈ Xk\Xk′ , k 6= k′. M is half-

dominant if for all such states z and each k = A,B, there exists some action x′ ∈ Xk\Xk′

which is a strictly better response to z for k members than any action x ∈ XA ∩XB.

In any state z, as in the hypothesis of half-dominance, at most half of k′ members play action

x ∈ XA∩XB. Hence the expected payoff to a k member from playing x is at most 12θk + δkx.

Some action x′ ∈ Xk\Xk′ will be a strictly better response if dk >12θk + δkx or Dkx >

12.

This holds for all x ∈ XA ∩XB and k ∈ {A,B} if and only if

minx∈XA∩XB

mink∈{A,B}

Dkx >12. (5)

Remark 2. The condition under which miscoordination is stochastically stable, the strict

risk dominance condition (2), is weaker than half-dominance, defined by condition (5).

To see this, first note that:

DAx +DBx ≥ 2 minx∈XA∩XB

mink∈{A,B}

Dkx,

17In Kajii and Morris’ terminology, what we define is a ( 12 ,

12 )-dominant equilibrium.

14

which is greater than one by (5). Hence half-dominance implies strict risk dominance.

The converse is not true. Consider the following asymmetric example.

Example 2. Payoffs are θA = θB = θ, dA = d, dB = αd where α > 1, and δAx = δBx = 0 for

all x ∈ XA ∩XB.

We require θ < 2d for (5) to be satisfied and miscoordination to be half-dominant. However,

we only require θ < (1 + α)d for (2) to be satisfied and miscoordination to be stochastically

stable. To highlight the role of asymmetry parameterized here by α, recall that condition

ND requires that θ > αd. As α → θd, (1 + α)d→ d + θ. Hence when α is very large, (2) is

satisfied for almost all d > 0, whereas half-dominance is only satisfied for d > θ2.

While this section has focused on miscoordination, Proposition 2(ii) enables us to say some-

thing about the conditions under which a state of coordination is stochastically stable. Notice

that payoffs are simple in Example 2. Hence the states of coordination are stochastically

stable if and only if θ ≥ (1 + α)d. That is, coordination is the long run outcome when each

group’s coordination payoff is greater than or equal to the sum of miscoordination payoffs

across groups. More generally, one would expect that coordination payoffs need to be high

relative to the cultural biases against coordination for a state of coordination to prevail in

the long run.

Now it could be that when payoffs are not simple,M is uniquely stochastically stable despite

not being strictly risk dominant. This remains an open question. We will, however, derive a

(weaker) necessary and sufficient condition for M to be stochastically stable in section 3.2

under a different error structure, without requiring simple payoffs.

3 An Application and Extension

This section analyzes an application of our model to identity-based conflict and subsequently

extends the model to allow for occasional violation of cultural constraints.

15

3.1 Identity-Based Conflict

To illustrate how our model can be applied, consider the following application to identity-

based conflict inspired by Sen (2006). The setting will be rather stylized to keep the focus

on evolutionary pressures and how they shape identity formation rather than on precise

institutional details.

Suppose that individuals belonging to different cultural groups, A and B, interact in pairs.

They each choose one of L ≥ 3 identities x ∈ X and in doing so are bound by cultural

constraints. Let identities 1 and L be exclusive, clearly defining an ingroup and outgroup

along ethnic or religious lines. Identity 1 is an exclusive group A identity and identity L

is an exclusive group B identity. All other identities are inclusive; they are not associated

with an ethnicity or religion and may even emphasize membership in some common social

category (e.g. nation).18

Identity choice is constrained in the following manner: a member of group A would not

consider adopting the exclusive group B identity, XA = X\{L}. Likewise, a group B

member would not consider adopting the exclusive group A identity, XB = X\{1}. To

be concrete, an Egyptian Muslim may grow a beard and become an active member of the

Muslim Brotherhood—an exclusive Muslim identity. An Egyptian Christian may tattoo a

Coptic cross on his wrist and become an active member of the Coptic church—an exclusive

Christian identity. Alternatively, they could both choose to adopt a neutral Egyptian identity

without any clear religious markers. What an Egyptian Muslim would not ordinarily do is

consider adopting an exclusive Christian identity and vice versa.

Given these cultural constraints, we shall show that exclusive ethnic or religious identities

can persist even when they are Pareto dominated by coordination on an inclusive identity.

Let us specify payoffs. Suppose that players obtain privileged access to group resources worth

β if they adopt an exclusive identity in intergroup interactions. Iannaccone (1992, 1994)

demonstrate how religious groups that require stigmatizing modes of dress and behavior can

more efficiently produce religious club goods, including social welfare services. For example,

growing a beard and joining the Muslim Brotherhood can give one preferential access to

group-provided healthcare services and employment opportunities; it can also confer higher

18Note that individuals are born into either group A or B. This is not a choice. However, they can choosewhether to emphasize their exclusive group identity or adopt an inclusive identity.

16

status within one’s group (e.g. Wickham 2002). There are, however, even stronger incentives

to coordinate in our model, as coordination on an inclusive identity limits the chance of

conflict between members of different cultural groups.

Each individual has property worth w. If the two players in an interaction choose the same

identity, then no conflict takes place and each individual retains w. If their identities differ,

then a violent winner-takes-all contest ensues in which each individual’s property is up for

grabs. The total prize in this contest, worth 2w, is allocated based on a standard Tullock

contest.19 The probability that the group A member wins the prize when exerting eA units

of effort in the contest is eA/(eA + eB), while the cost of exerting eA units of effort is simply

eA. The expected payoff to the group A member is thus:

eAeA + eB

2w − eA. (6)

The expected payoff to individual B is given by switching the A and B subscripts. The

unique Nash equilibrium contest payoff for each individual is 12w.

Thus the payoff from choosing an inclusive identity x ∈ {2, . . . L − 1} is I(x, xj)w + (1 −I(x, xj))12w, where xj is the action chosen by the player with whom the individual is matched.

That is, if coordination on an inclusive identity occurs, I(x, x) = 1, individuals retain their

existing wealth w. Otherwise, conflict ensues and each receives an expected contest payoff of12w. The payoff from choosing an exclusive identity is 1

2w + β, since conflict occurs for sure.

This is equivalent to our model of coordination with coordination payoffs θA = θB = 12w and

cultural biases dA − δAx = dB − δBx = β.

Notice that payoffs are simple, meaning uniform over all inclusive (mutually permissible)

identities x ∈ XA ∩XB. According to Proposition 2(ii) then, condition (2) is necessary and

sufficient for miscoordination to be stochastically stable. That is,M is stochastically stable

if and only if β/θA + β/θB > 1. Define the critical level of cultural bias, denoted by β, as

follows:

β

(1

θA+

1

θB

)= 1. (7)

Miscoordination is uniquely stochastically stable (for n sufficiently large) if and only if β > β.

Substituting θA = θB = 12w and manipulating (7), we find that β = 1

4w. That is to say,

19This contest function was first applied to conflict by Skaperdas (1992) and intergroup conflict by McBrideet al. (2011) and Sambanis & Shayo (2013). Sambanis and Shayo suggest that lower levels of ethnic conflictare observed when a common national identity is adopted, because individuals who adopt an ethnic identityaim to maximize the share of national resources captured by their ethnic group.

17

exclusive identities will be adopted virtually all of the time if the value of access to cultural

group resources, β, is greater than one quarter of an individual’s wealth. Holding the value

of group resources constant, one could predict on this basis that poorer societies are ‘more

likely’ to end up miscoordinating.20,21

Now consider a mean-preserving wealth spread, where wA = w + ∆ and wB = w − ∆.

Obviously, this can be generated by transferring resources from one group to the other.

Coordination payoffs are now:

θA = w + ∆− 12w = 1

2w + ∆

θB = w −∆− 12w = 1

2w −∆. (8)

Note that in the event of miscoordination, the expected payoff in the contest is still 12w for

each individual, since the total prize (2w) remains the same.

We show that the critical level of cultural bias is decreasing in ∆. That is, more unequal

societies are ‘more likely’ to end up miscoordinating. The reason for this is not obvious.

Notice that inequality does not alter either agent’s equilibrium contest payoff of 12w. Hence

this is not a matter of the poorer group having an incentive to exert greater expropriation

effort. The mean-preserving wealth spread does however translate into a mean-preserving

spread in coordination payoffs [see (8)]. It turns out that the effect of this change is asym-

metric, in the following sense. Fewer errors by A members are now required for an exclusive

identity to be a best response for poorer B members. On the other hand, a larger number of

errors by B members is required for an exclusive identity to be a best response for wealthier

A members. That is, poorer individuals become more inclined to choose exclusive identities,

while wealthier individuals persist more strongly in coordination. The former dominates,

however, and the net effect is to increase the stability of miscoordination. Hence inequality

can translate into ethnic/religious conflict, not for the usual reasons related to incentives for

expropriation, but due to out-of-equilibrium selection pressures.

20Of course, the value of access to group resources β may depend on w. But we have reason to thinkthat this effect is weak. Production of club goods typically involves labor-intensive modes of production(Iannaccone 1992, Berman 2000), especially in poorer societies (see Wickham 2002).

21Fearon & Laitin (2003) report a large and significant negative relationship between income levels andthe incidence of civil war. Their explanation centers on the link between higher income levels and theopportunity cost of insurgency. More closely related to our explanation is work by McBride et al. (2011).They propose that conflict destroys resources. Hence in a repeated games setting, higher income levelsprovide greater incentive for groups to strike a peaceful agreement.

18

We can state the following proposition:

Proposition 3 Consider the example of identity-based conflict with wealth levels wA = w+∆

and wB = w −∆, where ∆ ∈ [0, 12w). The critical level of cultural bias is:

(i) strictly increasing in w,

(ii) strictly decreasing in ∆.

The conclusion from this stylized example is that exclusive ethnic and religious identities are

more persistent in poorer, more unequal societies.

In related work on 2× 2 coordination games by Quilter (2007) and Neary (2012), members

of two groups play a single action against all members of the population. A different form

of miscoordination can arise in their setting, due to a tradeoff between ingroup coordination

on one’s preferred action and coordination with outgroup members. Our analysis points

to a different, but not mutually exclusive, source of miscoordination based on interactions

between groups. Indeed there is reason to believe that identity choice may be driven primarily

by its effect on outgroup interactions. A central theme in the economics of religion literature

is that exclusive identities are designed as a ‘tax’ on outgroup activity (see Iannaccone 1992,

Berman 2000). For example, Aksoy & Gambetta (2015) show that contact with natives

in Belgium increases the propensity of highly-religious Muslim women to veil. See also

Douglas (1966) on how taboos maintain community boundaries. In addition, our analysis

links miscoordination to economically relevant variables such as poverty and inequality, and

applies when individuals can choose different actions when interacting with ingroup and

outgroup members.

3.2 Violating Cultural Constraints

Consider two compatible but misaligned groups. We have established that differing cultural

constraints can lead to miscoordination among such groups. Does this result hinge upon

strict observance of cultural constraints? Does occasional violation of cultural constraints

destabilize miscoordination?

19

We shall address this question in the following manner. Recall that introducing small errors

into the strategy revision process enables us to make sharp predictions about the evolution

of play in the long run. We have hitherto assumed that individuals do not violate cultural

constraints when making errors. An error by a revising group k member has so far consisted

of a strategy chosen at random from Xk. We shall now allow errors to be unconstrained.

With probability ε, a revising player chooses an action at random from the global action

set X. (Note that (1) specifies payoffs even for x /∈ Xk.) This means that individuals can

violate cultural constraints, but the likelihood of doing so vanishes at the same rate as regular

nonbest responses.22 It turns out that allowing players to violate cultural constraints in this

way leads to sharper results and strengthens the stability of miscoordination M.

Another interpretation of this specification is that all players can choose all actions in X,

but actions in X\Xk are strictly dominated for k members: dk > θk + δkx for x ∈ X\Xk.

Hence such actions are only chosen erroneously. Kim & Wong (2010) show that any recurrent

class can be made stochastically stable by introducing some dominated strategies. Adding

dominated strategies in the manner described here, however, only strengthens the stability

of miscoordination.

We can now state a necessary and sufficient condition for M to be stochastically stable,

without imposing restrictions such as simple payoffs.

Proposition 4 Suppose the groups are compatible but misaligned. Consider the perturbed

best response dynamic with unconstrained errors. For n sufficiently large, the unique stochas-

tically stable class is miscoordination M if and only if

minx∈XA∩XB

(2 mink∈{A,B}

Dkx + maxk∈{A,B}

Dkx

)> 1. (9)

By inspection, condition (9) is weaker than strict risk dominance (2). Let us reconsider

Example 2 for a moment. Under constrained errors, we learned that the states of coordination

are stochastically stable if and only if θ ≥ (1 +α)d. Under unconstrained errors, (9) implies

that the states of coordination are stochastically stable if and only if θ ≥ (2 + α)d. For

22A non-vanishing likelihood of violating cultural constraints is analytically equivalent to allowing indi-viduals to revise their cultural constraints.

20

coordination to be stochastically stable, coordination payoffs still need to be high relative to

cultural biases against coordination. Occasional violation of cultural constraints raises the

threshold, making coordination harder to achieve.

The intuition for this result is as follows. When errors are unconstrained, members of one

group can make a sequence of errors in which they violate cultural constraints and choose

a cultural ideal of the other group. At this point, not only can a revising player gain from

choosing his cultural ideal as before, but there is also some chance that he coordinates on

his own cultural ideal with a deviant player from the other group. Hence introducing such

errors can make it easier to break from a state of coordination and can thereby strengthen

miscoordination. Of course, large-scale violations of cultural constraints may ultimately

undermine the force of such constraints, a situation examined elsewhere by Fershtman et al.

(2011).

4 Concluding Comments

This paper studies interactions between members of two cultural groups who are matched

to play a coordination game with an arbitrary number of actions. When individuals ob-

serve cultural constraints, miscoordination between cultural groups is a surprisingly stable

phenomenon. Occasional violation of cultural constraints can further strengthen miscoor-

dination. Unlike prior work, the type of miscoordination uncovered here occurs even when

there is no tradeoff between ingroup and outgroup coordination. Yet, since externalities are

limited in our model, it is possible that we have understated the likelihood of miscoordi-

nation. For example, we have only considered payoffs from adopting an inclusive identity

which are linear in the number of outgroup members who adopt an inclusive identity. There

are, however, cases in which only a few salient deviants are needed to create general distrust

between ethnic groups, communal riots and civil war (e.g. de Figueiredo & Weingast 1999).

Such externalities would compound the inefficiencies uncovered in this paper.

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Appendix

Proof of Proposition 1. We shall characterize the set of recurrent classes of the best

response dynamic. As any finite Markov chain converges almost surely to one of its recurrent

classes, this establishes the proposition.

Firstly, we claim that there are no recurrent classes of the best response dynamic other

than the states of coordination andM. To establish the claim it suffices to show that there

is a positive probability of transiting from any state to a state of coordination or a state in

M in a finite number of periods.

Consider an arbitrary time t and state zt. Let x be a best response for group A members.

Suppose that in period t, the set of revising players is Rt = NA and that each of them chooses

x. In addition, suppose that in period t + 1, Rt+1 = NB. All of this occurs with positive

probability.

Case 1. x ∈ XA ∩XB. (By definition, this is impossible if the cultures are incompatible.)

Then the expected payoff to each revising group B member from playing x is θBx + δBx for

all samples. The payoff from choosing x′ 6= x is at most dB, which is less than the payoff

from x by ND. Hence each revising group B member chooses x and the dynamic transits

with probability to a state of coordination in two periods.

Case 2. x /∈ XA ∩ XB. Then the payoff to each revising group B member from playing

any x′ ∈ XB is δBx′ . Hence each revising group B member chooses an action in XB ≡argmaxx∈XB

δBx. When the cultures are misaligned, every such action lies outside XA ∩XB,

as does x. Hence the dynamic transits with positive probability to a state in M in two

periods. When the cultures are aligned, we have either XA ∩XB is nonempty or XB ∩XA

is nonempty. Thus the dynamic transits with positive probability in at most two periods

to a state in which each group B member plays an action x′ ∈ XA ∩ XB. From there, the

argument used in case 1 establishes that the dynamic transits to a state of coordination in

finite time.

This establishes the claim.

The fact that the states of coordination are recurrent classes of the best response dynamic

if and only if the cultures are compatible follows immediately from the argument in case 1.

That M is a recurrent class if and only if the cultures are misaligned follows immediately

from the argument in case 2. This establishes the proposition. �

Proposition 2. The proof of Proposition 2 employs two tree-surgery arguments. These ar-

guments make use of the following three lemmas and corollary, which compare the resistance

of transitions between recurrent classes, defined as follows. The cost of a path (z1, z2, . . . zH)

is the minimum number of errors required for the perturbed dynamic (ε > 0) to transit

26

from state z1 to zH along this path. The resistance r(z1, zH) is the minimum cost of a path

starting at z1 and ending in zH . The definition can be extended so that for sets E and E ′,

the resistance r(E,E ′) is the minimum cost of a path starting in some state in E and ending

in some state in E ′.

In addition, let xi(z) be the action played by i in state z. A direct path (z1, z2, . . . zH) from

state z1 to zH is one in which xi(zh) ∈ {xi(z1), xi(zH)} for all h = 1, 2, . . . H and i ∈ N .

Define Xi(E) ≡ {xi(z) : z ∈ E}. A direct path (z1, z2, . . . zH) from set E to E ′ is one in

which z1 ∈ E, zH ∈ E ′ and xi(zh) ∈ Xi(E) ∪Xi(E

′) for all h = 1, 2, . . . H and i ∈ N .

Lemma 1 Let zx denote the state of coordination on x. For all distinct pairs (x, x′) ∈(XA ∩XB)2,

r(M, zx) ≤ r(zx′ , zx).

Proof. Let r(zx′ , zx) = a. This means that for some k ∈ {A,B}, after a errors by k′

members, x is a best response for k members, and hence a weakly better response than a

mutually impermissible action. That is:

dk ≤a

nθk + δkx. (10)

Let αk be the minimum number of errors such that (10) holds:

αk =

⌈dk − δkxθk

n

⌉≡ dDkxne. (11)

We know r(zx′ , zx) = a ≥ mink∈{A,B} αk. By inspection of (10), mink∈{A,B} αk is the

minimum cost of a direct path from any state inM to zx. Hence r(M, zx) ≤ mink∈{A,B} αk.

This establishes the Lemma. �

Lemma 2 Suppose that in state z, na members of group A and nb members of group B

use strategy x. Define z′ as the state in which the same is true, with the remaining n − namembers of group A and n− nb members of group B using mutually impermissible actions.

Then r(z′, zx) ≤ r(z, zx).

Proof. The argument in the proof of Lemma 1 establishes the result. �

Lemma 3 If miscoordination M is strictly risk dominant, then for n sufficiently large

r(zx,M) < r(M, zx)

for all x ∈ XA ∩XB.

27

Proof. We claim that a direct path fromM to zx has minimum cost among all paths from

M to zx. Consider an arbitrary indirect path from M to zx. Since the path is indirect, it

runs through a state z in which some mutually permissible action x′ 6= x is played. The

minimum cost of such a path is r(M, z) + r(z, zx).

In state z, replace every action by a k member not equal to x with an action in Xk\Xk′ ,

k′ 6= k. Label this state z′. There is a direct path from M to zx through state z′. The

minimum cost of such a path is r(M, z′) + r(z′, zx). Clearly, r(M, z′) < r(M, z). In

addition, r(z′, zx) ≤ r(z, zx) by Lemma 2. Hence the direct path has lower cost than the

indirect path, establishing the claim.

By (11), the minimum cost of a direct path is mink∈{A,B}dDkxne. Hence we need only find

a path from zx toM with lower cost. Consider a direct path. Starting from zx, k′ members

make a errors. For x′′ ∈ Xk\Xk′ to be a best response for k members and to thereby reach

a state in M without further errors, a must satisfy(1− a

n

)θk + δkx ≤ dk.

That is,

a ≥(

1− dk − δkxθk

)n ≡ (1−Dkx)n.

Thus the minimum cost of such a transition is

mink∈{A,B}

⌈(1−Dkx)n

⌉.

For n sufficiently large, this is less than mink∈{A,B}dDkxne if and only if

mink∈{A,B}

Dkx > mink∈{A,B}

(1−Dkx

)min

k∈{A,B}Dkx + max

k∈{A,B}Dkx > 1

DAx +DBx > 1. (12)

The last inequality holds, because M strictly risk dominates zx [see (2)]. �

The next result follows immediately from Lemmas 1 and 3:

Corollary 1 If miscoordination M is strictly risk dominant, then for n sufficiently large

r(zx,M) < r(zx′ , zx)

for all distinct pairs (x, x′) ∈ (XA ∩XB)2.

28

𝑧𝐻

M

…

𝑧1

𝑧0

𝑧𝐻−1

(a) 0-tree

𝑧𝐻

M

…

𝑧1

𝑧0

𝑧𝐻−1

(b) M-tree

Figure 1: Beginning with the 0-tree in (a), steps 1-2 produce the M-tree in (b).

Proof of Proposition 2. (i) Suppose thatM is strictly risk dominant. We shall show that

for n sufficiently large, M is the unique stochastically stable class using the spanning-tree

method of Young (1993a).

The following tree-surgery argument is used. An x-tree is a tree consisting of a set of

|XA ∩ XB| directed edges, one emanating from each recurrent class (node) other than zx.

Consider a tree, say rooted at 0 and denoted by T0, that has minimum resistance among x-

trees. Through a series of operations on T0 we will produce anM-tree with lower resistance

than T0. Hence all trees with minimum resistance areM-trees. By Young (1993a, Theorem

2) then, M is the unique stochastically stable class.

Again let T0 have minimum resistance among x-trees. By definition, there is a unique route

fromM to z0 through T0. (See Figure 1(a) for example.) Denote the nodes on this route by

zH , zH−1, . . . z1, z0. The edges composing this route are (M, zH), (zH , zH−1), . . . (z1, z0).

Step 1. Replace edge (M, zH) with (zH ,M).

Step 2. Replace each edge (zh, zh−1) with (zh−1,M) for h = 1, . . . H.

Steps 1-2 produce an M-tree. (See Figure 1(b) for example.) The difference in resistance

between the original 0-tree and this M-tree is:

[r(M, zH)− r(zH ,M)

]+

H∑h=1

[r(zh, zh−1)− r(zh−1,M)

]. (13)

For n sufficiently large, the first term is positive by Lemma 3 and the second term, if

it exists, is positive by Corollary 1. Hence the M-tree produced by steps 1-2 has lower

29

𝑧𝑥

M

(a) M-tree

𝑧𝑥

M

(b) x-tree

Figure 2: Beginning with the M-tree in (a), replace edge (zx,M) with (M, zx) to pro-duce the x-tree in (b).

resistance than the original x-tree, T0. Since the original x-tree was chosen arbitrarily, all

trees with minimum resistance are M-trees, establishing the result.

(ii) Again a tree-surgery argument is employed. Consider an arbitrary M-tree. There

exists at least one edge (zx,M), for some x ∈ XA ∩ XB. Replace this edge with (M, zx).

This transforms the original M-tree into an x-tree. See Figure 2 for an illustration.

The difference in resistance between the original M-tree and this x-tree is:

r(zx,M)− r(M, zx).

We claim that this is nonnegative when payoffs are simple and M is not strictly risk

dominant. Hence for every M-tree there exists an x-tree with no greater resistance. By

Young (1993a, Theorem 2) then, M is not the unique stochastically stable class.

Let us now establish the claim. In the proof of Lemma 3, we showed that r(M, zx) =

mink∈{A,B}dDkxne and the cost of a direct path from zx toM equals mink∈{A,B}d(1−Dkx)ne.

An indirect path from zx to M involves at least b erroneous plays of some mutually

permissible action x′ 6= x by k′ members, so that x′ is a best response for k members. Hence

b satisfies: (1− b

n

)θk + δkx ≤

b

nθk + δkx′

2b

nθk ≥ θk + δkx − δkx′

b ≥ 1

2n.

The last line utilizes the fact that δkx = δkx′ because payoffs are simple. Thus the cost of

an indirect path from zx to M is at least d12ne.

30

Hence when payoffs are simple, r(zx,M) ≥ r(M, zx) if n is sufficiently large and

min

{min

k∈{A,B}(1−Dkx),

1

2

}≥ min

k∈{A,B}Dkx. (14)

We shall now show that (14) is satisfied. By hypothesis, M is not strictly risk dominant.

As payoffs are simple, this means that M is risk dominated by all states of coordination.

By (12) this is equivalent to

mink∈{A,B}

Dkx ≤ mink∈{A,B}

(1−Dkx

), (15)

which implies

mink∈{A,B}

Dkx ≤ maxk∈{A,B}

(1−Dkx

)min

k∈{A,B}Dkx ≤ 1− min

k∈{A,B}Dkx

mink∈{A,B}

Dkx ≤ 12. (16)

Together (15) and (16) imply that (14) holds. This establishes the claim and indeed part

(ii) of the proposition. �

Proof of Proposition 3. The result follows immediately from the fact that 1/x is strictly

positive, strictly decreasing, and strictly convex. �

Proposition 4. When ε = 0, there is no violation of cultural constraints. Hence the

recurrent classes of the best response dynamic (ε = 0) are still given by Proposition 1.

These are the candidates for stochastic stability.

The only change in computation of resistances is to transitions from states of coordination

to M. Hence Lemmas 1 and 2 still apply.

We shall now introduce two new lemmas and a corollary. Lemma 4 and Corollary 2 are

analogs of Lemma 3 and Corollary 1 respectively for the case of unconstrained errors. We

shall then employ the tree-surgery arguments used in Proposition 2 to establish the result.

Lemma 4 If (9) is satisfied, then for n sufficiently large

r(zx,M) < r(M, zx)

for all x ∈ XA ∩XB.

31

Proof. Recall that r(M, zx) = mink∈{A,B}dDkxne.

Let us now compute r(zx,M). On the minimum cost path from zx toM one of two things

must occur. Starting from zx, through a series of errors, either (i) a mutually impermissible

action becomes a best response, in which case the process transits toM without any further

errors, or (ii) an action x′ ∈ XA ∩XB becomes a best response.

The minimum cost path conforming to case (i) is a direct path as follows. As errors are

unconstrained, a transition from zx to M can occur with a erroneous plays of an action in

Xk \Xk′ by k′ members, where a satisfies(1− a

n

)θk + δkx ≤

a

nθk + dk. (17)

The value of a that equates (17) is (1−Dkx)n2. Hence the cost of such a path from zx to

M is

mink∈{A,B}

⌈(1−Dkx)

n

2

⌉= min

k∈{A,B}

⌈(1− dk − δkx

θk

)n

2

⌉. (18)

The minimum cost path conforming to case (ii) involves at least b erroneous plays of

some mutually permissible action x′ 6= x by k′ members, so that x′ is a best response for k

members. Hence b satisfies (1− b

n

)θk + δkx ≤

b

nθk + δkx′ .

Thus the cost of such a path from zx to M is at least

mink∈{A,B}

⌈(1− δkx′ − δkx

θk

)n

2

⌉. (19)

This is greater than (18) because dk > δkx′ by misalignment. Hence the minimum cost path

from zx to M is direct and r(zx,M) equals (18).

Therefore, when errors are unconstrained and n is sufficiently large the following statements

are equivalent:

r(M, zx) > r(zx,M),

mink∈{A,B}

Dkx > mink∈{A,B}

12

(1−Dkx

),

2 mink∈{A,B}

Dkx + maxk∈{A,B}

Dkx > 1. (20)

The last line holds by hypothesis (9). �

The next result follows immediately from Lemmas 1 and 4.

32

Corollary 2 If (9) is satisfied, then for n sufficiently large

r(zx,M) < r(zx′ , zx)

for all distinct pairs (x, x′) ∈ (XA ∩XB)2.

Lemma 5 r(zx,M) ≤ r(zx, zx′) for all distinct (x, x′) ∈ (XA ∩XB)2.

Recall that r(zx,M) equals (18).

Now consider r(zx, zx′). On the minimum cost path from zx to zx′ one of two things must

occur. Starting from zx, through a series of errors, either (i) a mutually impermissible action

becomes a best response, in which case r(zx, zx′) ≥ r(zx,M), or (ii) an action x′′ ∈ XA∩XB

(possibly x′) becomes a best response. The path with minimum cost that conforms to the

case (ii) involves β erroneous plays of x′′ by k′ members, where β equals (19).

We established in the proof of Lemma 4 that (19) is greater than (18). This establishes

the Lemma. �

Proof of Proposition 4. To prove that (9) is sufficient for M to be stochastically stable

we use the same tree-surgery argument as in Proposition 2(i). Starting with an arbitrary

x-tree, we transform it through steps 1-2 into an M-tree. Again the difference in resistance

between the original x-tree and the resultantM-tree is given by (13). If (9) holds, then the

first term is positive by Lemma 4 and the second term, if it exists, is positive by Corollary

2. Hence (9) is sufficient.

To establish that (9) is also necessary, note that Lemma 5 implies that the minimum cost

M-tree is composed entirely of direct links, i.e. edges (zx,M) for each x ∈ XA ∩XB.

Suppose that (9) were violated, so that 2 mink∈{A,B}Dkx + maxk∈{A,B}Dkx ≤ 1 for some

x ∈ XA ∩ XB. Recall that the minimum cost M-tree contains edge (zx,M). Replace

this edge with (M, zx). This operation transforms the minimum cost M-tree into an x-

tree. The difference in resistance between the original M-tree and the resultant x-tree is

r(zx,M)− r(M, zx) which is non-negative as (9) is violated [see (20)]. Hence M is not the

unique stochastically stable class. �

33